$\tan x-\cot x=6\Rightarrow \frac{\sin x}{\cos x}-\frac{\cos x}{\sin x}=6$ $\Rightarrow \frac{{{\sin }^{2}}x-{{\cos }^{2}}x}{\sin x\cos x}=6$ از آنجایی که $2sin x cos x=sin2x$ و ${{\cos }^{2}}x-{{\sin }^{2}}x=\cos 2x$ داریم: $\Rightarrow \frac{-\cos 2x}{\frac{1}{2}\sin 2x}=6\Rightarrow -\tan 2x=\frac{1}{3}\Rightarrow \tan 2x=-\frac{1}{3}$